Quantum Machine Learning For Classical Data Leonard P. Wossnig A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. arXiv:2105.03684v2 [quant-ph] 12 May 2021 Department of Computer Science University College London May 13, 2021 2 I, Leonard P. Wossnig, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the work. Abstract In this dissertation, we study the intersection of quantum computing and supervised machine learning algorithms, which means that we investigate quantum algorithms for supervised machine learning that operate on classical data. This area of re- search falls under the umbrella of quantum machine learning, a research area of computer science which has recently received wide attention. In particular, we in- vestigate to what extent quantum computers can be used to accelerate supervised machine learning algorithms. The aim of this is to develop a clear understanding of the promises and limitations of the current state-of-the-art of quantum algorithms for supervised machine learning, but also to define directions for future research in this exciting field. We start by looking at supervised quantum machine learning (QML) algorithms through the lens of statistical learning theory. In this frame- work, we derive novel bounds on the computational complexities of a large set of supervised QML algorithms under the requirement of optimal learning rates. Next, we give a new bound for Hamiltonian simulation of dense Hamiltonians, a major subroutine of most known supervised QML algorithms, and then derive a classical algorithm with nearly the same complexity. We then draw the parallels to recent ‘quantum-inspired’ results, and will explain the implications of these results for quantum machine learning applications. Looking for areas which might bear larger advantages for QML algorithms, we finally propose a novel algorithm for Quantum Boltzmann machines, and argue that quantum algorithms for quantum data are one of the most promising applications for QML with potentially exponential advantage over classical approaches. Acknowledgements I want to thank foremost my supervisor and friend Simone Severini, who has always given me the freedom to pursue any direction I found interesting and promising, and has served me as a guide through most of my PhD. Next, I want to thank Aram Harrow, my secondary advisor, who has always been readily available to answer my questions and discuss a variety of research topics with me. I also want to thank Carlo Ciliberto, Nathan Wiebe, and Patrick Rebentrost, who have worked closely with me and have also taught me most of the mathematical tricks and methods upon which my thesis is built. I furthermore want to thank all my collaborators throughout the years. These are in particular Chunhao Wang, Andrea Rocchetto, Marcello Benedetti, Alessan- dro Rudi, Raban Iten, Mark Herbster, Massimiliano Pontil, Maria Schuld, Zhikuan Zhao, Anupam Prakash, Shuxiang Cao, Hongxiang Chen, Shashanka Ubaru, Haim Avron, and Ivan Rungger. Almost in an equal contribution, I also want to thank Fernando Brandao,˜ Youssef Mroueh, Guang Hao Low, Robin Kothari, Yuan Su, Tongyang Li, Ewin Tang, Kanav Setia, Matthias Troyer, and Damian Steiger for many helpful discussions, feedback, and enlightening explanations. I am particularly grateful to Edward Grant, Miriam Cha, and Ian Horobin, who made it possible for me to write this thesis. I want to acknowledge UCL for giving me the opportunity to pursue this PhD thesis, and acknowledge the kind support of align Royal Society Research grant and the Google PhD Fellowship, which gave me the freedom to work on these interesting topics. Acknowledgements 5 Portions of the work that are included in this thesis were completed while I was visiting the Institut Henri Poincare´ of the Sorbonne University in Paris. I particularly want to thank Riam Kim-McLeod for the support and help with the editing of the thesis. I finally want to thank my family for the continued love and support. Impact Statement Quantum machine learning bears promises for many areas, ranging from the health- care to the financial industry. In today’s world, where data is available in abundance, only novel algorithms and approaches are enabling us to make reliable predictions that can enhance our life, productivity, or wealth. While Moore’s law is coming to an end, novel computational paradigms are sought after to enable a further growth of processing power. Quantum computing has become one of the prominent can- didates, and is maturing rapidly. The here presented PhD thesis develops and stud- ies this novel computational paradigm in light of existing classical solutions and thereby develops a path towards quantum algorithms that can outperform classical approaches. Contents 1 Introduction and Overview 12 1.1 Synopsis of the thesis . 15 1.2 Summary of our contributions . 16 1.3 Statement of authorship . 20 2 Notation And Mathematical Preliminaries 23 2.1 Notation . 23 2.2 Matrix functional analysis . 24 3 Statistical Learning Theory 29 3.1 Review of key results in Learning Theory . 31 3.1.1 Supervised Learning . 32 3.1.2 Empirical risk minimization and learning rates . 33 3.1.3 Regularisation and modern approaches . 37 3.2 Review of supervised quantum machine learning algorithms . 39 3.2.1 Recap: Quantum Linear Regression and Least Squares . 40 3.2.2 Recap: Quantum Support Vector Machine . 43 3.3 Analysis of quantum machine learning algorithms . 44 3.3.1 Bound on the optimisation error . 46 3.3.2 Bounds on the sampling error . 49 3.3.3 Bounds on the condition number . 50 3.4 Analysis of supervised QML algorithms . 57 3.5 Conclusion . 58 Contents 8 4 randomised Numerical Linear Algebra 60 4.1 Introduction . 61 4.2 Memory models and memory access . 62 4.2.1 The pass efficient model . 62 4.2.2 Quantum random access memory . 64 4.2.3 Quantum inspired memory structures . 67 4.3 Basic matrix multiplication . 69 4.4 Hamiltonian Simulation . 74 4.4.1 Introduction . 75 4.4.2 Related work . 80 4.4.3 Applications . 85 4.4.4 Hamiltonian Simulation for dense matrices . 87 4.4.5 Hamiltonian Simulation with the Nystrom¨ method . 102 4.4.6 Beyond the Nystrom¨ method . 126 4.4.7 Conclusion . 126 5 Promising avenues for QML 130 5.1 Generative quantum machine learning . 131 5.1.1 Related work . 132 5.2 Boltzmann and quantum Boltzmann machines . 133 5.3 Training quantum Boltzmann machines . 135 5.3.1 Variational training for restricted Hamiltonians . 137 5.3.2 Gradient based training for general Hamiltonians . 143 5.4 Conclusion . 147 6 Conclusions 150 Appendices 156 A Appendix 1: Quantum Subroutines 156 A.1 Amplitude estimation . 156 A.2 The Hadamard test . 157 Contents 9 B Appendix 2: Deferred proofs 159 B.1 Derivation of the variational bound . 159 B.2 Gradient estimation . 161 B.2.1 Operationalizing the gradient based training . 163 B.3 Approach 2: Divided Differences . 170 B.3.1 Operationalising . 187 Bibliography 192 List of Figures 1.1 Different fields of study in Quantum Machine Learning. The dif- ferent areas are related to the choice of algorithm, i.e., whether it is executed on a quantum or classical computer, and the choice of the target problem, i.e., whether it operates on quantum or classical data. 13 3.1 Summary of time complexities for training and testing of differ- ent classical and quantum algorithms when statistical guarantees are taken into account. We omit polylog(n;d) dependencies for the p quantum algorithms. We assume e = Q(1= n) and count the ef- fects of measurement errors. The acronyms in the table refer to: least square support vector machines (LS-SVM), kernel ridge re- gression (KRR), quantum kernel least squares (QKLS), quantum kernel linear regression (QKLR), and quantum support vector ma- chines (QSVM). Note that for quantum algorithms the state ob- tained after training cannot be maintained or copied and the algo- rithm must be retrained after each test round. This brings a factor proportional to the train time in the test time of quantum algorithms. Because the condition number may also depend on n and for quan- tum algorithms this dependency may be worse, the overall scaling of the quantum algorithms may be slower than the classical. 59 4.1 An example of the data structure that allows for efficient state prepa- ration using a logarithmic number of conditional rotations. 66 List of Figures 11 4.2 An example of the classical (dynamic) data structure that enables × efficient sample and query access for the example of H 2 C2 4... 69 p 4.3 Comparing our result O(t d kHk polylog(t;d;kHk;1=e)) with other quantum and classical algorithms for different models. Since the qRAM model is stronger than the sparse-access model and the classical sampling and query access model, we consider the advan- tage of our algorithm against others when they are directly applied to the qRAM model. 86 5.1 Comparison of previous training algorithms for quantum Boltz- mann machines. The models have a varying cost function (objec- tive), contain (are able to be trained with) hidden units, and have different input data (classical or quantum). 132 Chapter 1 Introduction and Overview In the last twenty years, due to increased computational power and the availability of vast amounts of data, machine learning (ML) has seen an immense success, with applications ranging from computer vision [1] to playing complex games such as the Atari series [2] or the traditional game of Go [3]. However, over the past few years, challenges have surfaced that threaten the end of this revolution.